Problem: $f(x, y) = 2y - xy$ We have a change of variables: $\begin{aligned} x &= X_1(u, v) = -2u - 5v \\ \\ y &= X_2(u, v) = 3u - 2v \end{aligned}$ What is $f(x, y)$ under the change of variables? Choose 1 answer: Choose 1 answer: (Choice A) A $8u^2 - 16v^2 + 13uv + 6u - 4v$ (Choice B) B $6u^2 - 10v^2 + 11uv + 6u - 4v$ (Choice C) C $-8u^2 + 16v^2 - 13uv + 6u - 4v$ (Choice D) D $-6u^2 + 10v^2 - 11uv + 6u - 4v$
Answer: When applying a change of variables, we substitute the new definition for $x$ and $y$ into the original equation. The original equation: $f(x, y) = 2y - xy$ Let's substitute $X_1(u, v)$ for $x$ and $X_2(u, v)$ for $y$. $\begin{aligned} f(x, y) &= 2(3u - 2v) - (-2u - 5v)(3u - 2v) \\ \\ &= 6u - 4v - (-6u^2 + 4uv - 15uv + 10v^2) \\ \\ &= 6u^2 - 10v^2 + 11uv + 6u - 4v \end{aligned}$ Therefore, under the change of variables, $f(x, y)$ becomes: $6u^2 - 10v^2 + 11uv + 6u - 4v$